MATH 212 Exam 1, Fall 2016

1866 #AUB as erage University ty of Beirut Quiz 1: MATH ¢ 212 (Introductory PDI 63) n aia Instructors: Sophie Moufawad & Wael Mahboub September 30, 2016 Duration: 70 minutes Name (Last, First): Student number: _ Section 1: 1lam-12pm (Sophie Moufawad) Section 2: 12pm-01pm (Wael Mahboub) Section 3: 0lpm-02pm (Wael Mahboub) For marker’s use only Problem Score 1 Qg /10 2 Lg /18 3 je /15 4 20 /20 5 | 35 /37 Total | 4¢/100 Y [10 points=7+3] Problem 1. Consider the following boundary value problem U+t3ue-u= 0, t>0, O<2r<3 (1) u(t,0) =~ e4, (2) u(t,3) = e%1, (3) (a) Find all the separable eigensolutions of the damped uniform transport equation (1). Let Uleit) = wit) viz) Ut=w'y £uUx=wv' subs in We $012? wy t 3Bwv!-wv =9 wiv= —-Zwv'twVv wiyew (-3vitv) (_ -2v'tV _ ee na Soy (wi-kw20 (4) _ayleve AV => -3v'4 (I-PA)V20 -- 6S) XelR (b) Find the separable eigensolutions that satisfy the boundary conditions (2), and (3). Udi)= et *® 2 e* a) " At -324 great ew =} g ulki3)= e - => | aca =) A=2 Mat AE -( =? a2 Se, ultir)= ee Page 2 of 10 [18 points] Problem 2. Solve the following initial value problem up + 4#uy=0, t>0, 2>0 u(0, 2) = ps. the PDE can be written % Ue + Ze, Ug =O He é Le dt” Yat (Ua dx oft dt iU- Htc 3 ce 4x? _t* (chatachishic cules ) Ef Zz => uaret(fer-£ ) lu is constanh afory the curve] (+z? fx) 2 fe ay —- A i a se -— Se, aie ng ( 4a [+x?-34% Page 3 of 10 4 [15 points] Problem 3. Solve the following initial value problem fie t>0 -- c=3 antian= 2 if 3<a<4 Lae 0, otherwise u,(0, 2) = 0. LAlemdeerl's. ucts FEt3+) +f4-3t) pL rau te A tay [ocerde in this Be, 9(*)=0 => uGjr= fvt3+) r $G-3t) ol Ll3t)= } 47 3Bdnt3td4 Oo polwix a f4-3t) = Zo? 3Lz-H< + 6) poWotwiee = f(1+34)= a J Fas Ayaan |? ? 33 2X4 BE 9o wi A in@: far3t) <fa-3t)-0 n@: flardt)=2 &F4-3+)=0 in@ : flet37 Paa)=t n@ : fatze)= fiu-3t)= 2 in: £6043) =tt-3t)=O i olve the following initial goal of this problem 1s to s [20 points= value problem. ugg = UT We T Vee t>0 u(0,£) = 0,veeR u(0,2) = 2 (a) Let v(t, 2) = e®u(t, 2). Show that v(t, 2) satisfies a wave equation, to be determined 1h uze V ure re Mie 2 OV -2E™ revs Unee= OY - re aa vy -2e Ve © or a =e Vi. a he ak ine PDE & Vee = He ep Prd © Va e "Vee = i Vane L tions. VGajee ug) = @ t(o)20 = F(*) - Vlart = & *y Ga) Fe * ol) O att palemberts : vdn)= P(urtde ee) i _l t | g(t) dt el att rt 2 z f ze dz Bee qt Ié - att a set = s Jee -e* ee at Oe ut t + ot fete” vee aa-tle +2 ‘ 14 Let [cute aoe rete’ Page 5 of 10 (c) Deduce u(t, x). uctit)= 6 Cv lit) PS # 4 € a I fatt)e*-G-bHe oat a8 ] ee 7 + f artyet-G-te te*- e*) a insight c lou [37 points=5+7+10+15] Problem 5. The goal of this problem is to solve the following initial value problem u(0,z) = 27, (4) Ut — Ute — 6Uee = sin(c +t), t>0, VeeR u,(0, x) = e* (a) Find a particular solution to the inhomogeneous pde uy — Ue — 6c = sin(x +t) of the form up(t, x) = cisin(x + t) + ccos(x +t), where 1,c2 ER. Up= qsin(at E) +2005 (xt +t) Upp = Gcos(att)-G@sin (utt) Upp, 2- C1 910 Cett)-C2@5 jett) UP = -G sin(xtt) C25 (xt+t) py = C\ cost rt) -@sin(xtt) UPye = C1 sin(etrt) -2 es (att) Gabe in the pde: . FE ~Girl at €) 16 snl) + gt) +asinlath) 6c, coslut t) = sulrtt) “Fig ee ee C8 Oy Wye ML BY , DW AM =-2Vy +3 $7 Wer Dh = BEE + OF oe f 1 Vig 2. AVp ap, 3 OM avn 29 Mee~ -2 Se ho 250 +3 ates ae Uys 2¥yop , AVY RN a MM 24 " “ay “5a * ate? UGLY = Vpp + 2¥pm + Vi = 2M Wy 2 aM oY 22M 2 = -2Voy +Voy + 3¥qy S Us in ‘ te dg: Of ah ha TR OO -25 Vpn} =@ = Vp =O Page 8 of 10 a a aq (34) ~° Vas ay B(1) Which ABO is come Favwchioal => VPN) = £8) + 9/0) whore ff) Fa) ave outers Sencon (son) =e) ) ay Ulert)= F/4-2t) t 9 (++3t) nN—_ oo insight c lu (c) Show that all the solutions to the homogeneous pde ue — tue — 6Uex = 0 have the same form as the solution found in part (b). (Hint: apply the change of variable from (t,x) to (,€), where u(t, x) = v(x, €).) [ ee) — insight => the colulion fo this pde is _ Ulbiat)}= fq-t) +9(nx3t) . dich 15 Abe gummaten of Pot py [idliour from bth Odes bifilar # bf[u]=?- (we tek the sammodion dive t bvear'ty!) Page 9 of 10